Ah, I think, I know why the Mengersponge has topological dimension of 1:

It *probably* has a representation similar to the Sierpinsky Sponge, by means of the 3D-Sierpinsky Arrowhead Curve:

Namely probably some sort of 3D version of the Sierpinsky Curve:

http://en.wikipedia.org/wiki/Sierpi%C5%84ski_curveBecause you can do this, you technically only need one parameter, so that's the topological dimension.

@taurus66:

The one I called "geometric" is the euclidean dimension.

My "topological" dimension is correct. The Menger Sponge still is a "three-dimensional" object in the sense of the euclidean dimension.

What was incorrect of me, was that the topological dimension has to be one less than the euclidean dimension. This is obviously not the case if you think of (topologically) one-dimensional curves that, to not be distorted by projections, may or may not require representations in R

^{n}, n>2, like for instance a helix that cannot exist in R

^{2} without distortion.

However, the Hausdorff dimension still is strictly greater than the topological dimension and smaller than the euclidean dimension.

Of course, my definition that you do need n parameters for n-dimensional topology becomes a bit weird at n=0 for a point cloud. However, it still seems accurate, since in that case, you simply can't desribe the set with a parameter. You have to explicitly list the points in some way, or give an algorithm that finds every point. - This algorithm might again use parameters but those are not of the same "kind" as usual geometric parameters.

My definitions are also failing as the euclidean dimension goes towards infinity. But so do most intuitively useful definitions, if not all...

The linked Lebesgue covering dimension is of course more general but for "usual" surface-sets that can also be reasonably ploted in eucildean geometry (and I guess, also in general geometry of varying curvature), my simplified ways of saying it should be equivalent.

I guess, I also exclude the case where you consider the volume, which should require one more parameter (for the before-mentioned sphere , if you do not fix r and vary it along intervals, you obviously obtain the respective ball or, if you go 0<a<=r<=b, a ball with a smaller ball taken out of it, with dimension (e.g. parameter-count) 3: r, phi, theta)

That *is* a gross simplification, but it's not difficult to rectify.

The topological dimension is essentially, for non obscure cases, the minimum number of parameters, you need to fully describe a set. This now holds for the volume case too.

"sufficiently nice" is "hinreichend nett", as in not a bastard topology. IMHO, the Menger Sponge is sufficiently nice, since it can be broken down in the above-mentioned curve that ultimately is made up of line segments.

Though I can not guarantee that for all fractals. Likely, there are some monster-toplogies that are not only monsters but also bastards.